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Multiband Homogenization of Metamaterials in Real-Space

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Multiband Homogenization of Metamaterials in Real-Space Submitted preprint. 1 Multiband Homogenization of Metamaterials in Real-Space: 2 Higher-Order Nonlocal Models and Scattering at External Surfaces 1, ∗ 2, y 1, 3, 4, z 3 Kshiteej Deshmukh, Timothy Breitzman, and Kaushik Dayal 1 4 Department of Civil and Environmental Engineering, Carnegie Mellon University 2 5 Air Force Research Laboratory 3 6 Center for Nonlinear Analysis, Department of Mathematical Sciences, Carnegie Mellon University 4 7 Department of Materials Science and Engineering, Carnegie Mellon University 8 (Dated: March 3, 2021) Dynamic homogenization of periodic metamaterials typically provides the dispersion relations as the end-point. This work goes further to invert the dispersion relation and develop the approximate macroscopic homogenized equation with constant coefficients posed in space and time. The homoge- nized equation can be used to solve initial-boundary-value problems posed on arbitrary non-periodic macroscale geometries with macroscopic heterogeneity, such as bodies composed of several different metamaterials or with external boundaries. First, considering a single band, the dispersion relation is approximated in terms of rational functions, enabling the inversion to real space. The homogenized equation contains strain gradients as well as spatial derivatives of the inertial term. Considering a boundary between a metamaterial and a homogeneous material, the higher-order space derivatives lead to additional continuity conditions. The higher-order homogenized equation and the continuity conditions provide predictions of wave scattering in 1-d and 2-d that match well with the exact fine-scale solution; compared to alternative approaches, they provide a single equation that is valid over a broad range of frequencies, are easy to apply, and are much faster to compute. Next, the setting of two bands with a bandgap is considered. The homogenized equation has also higher-order time derivatives. Notably, the homogenized model provides a single equation that is valid over both bands and the bandgap. The continuity conditions for the higher-order spatio-temporal homogenized equation are applied to wave scattering at a boundary, and show good agreement with the exact fine-scale solution. Using that the order of the highest time derivative is proportional to the number of bands considered, a nonlocal-in-time structure is conjectured for the homogenized equation in the limit of infinite bands, suggesting that homogenizing over finer length and time scales is a mechanism for the emergence of macroscopic spatial and temporal nonlocality. 9 1. Introduction 10 Metamaterials have the potential to display unusual properties, and have been the focus of much attention in + + 11 mechanics and electromagnetism, e.g. [DLT 21, LR18, Sd12, HHS07, MSGP 18, DB07, KK21, BVCV17, 12 GWWM06, SRGS09, HLR14, LS18, FTP15, HE21] and numerous others. A primary focus of much of this 13 effort is to find the dispersion relations in the context of infinite periodic systems, and then tailor or optimize 14 these relations to obtain desirable properties. 15 In this paper, we go beyond computing the dispersion relation to develop homogenized models, in the 16 context of linear periodic metamaterials. The homogenized models that we develop have two key features: 17 (1) they are posed in real space and time, i.e., not in frequency space; and (2) they are posed in terms 18 of a single equation that is valid over a large frequency range and across multiple bands. These features 19 can enable the application of the models to problems posed on complex geometries and with complex 20 time-dependent behavior, rather than infinite periodic systems at steady state and at specific frequencies. 21 Our approach is based on developing approximations to the dispersion relation and then inverting it to 22 obtain the real-space model. This requires us to balance between two competing objectives: (1) an accurate ∗ [email protected] y [email protected] z [email protected] 2 Figure 1. The metamaterial (left) gives a homogenized material with higher-order space-time derivatives (right), and consequent higher-order continuity conditions. 23 representation of the dispersion relation, and (2) tractability of the representation to inversion to real space 24 to obtain standard differential operators. For a single band, we find that a class of models that go back to 25 Toupin [Tou62] provides a good balance. In brief, this class of models can be considered as approximating 26 dispersion curves as rational functions; the inversion to real-space then provides standard but higher-order 27 differential operators. Specifically, the homogenized equation includes strain gradients as well as spatial 28 gradients of the inertial terms. When we extend this to higher bands, we find the appearance of higher-order 29 time derivatives as well. 30 To summarize our overall procedure, we: (1) start with a given dispersion relation that is derived 31 under assumptions of periodicity; (2) use a rational function approximation of the dispersion relation to 32 develop a homogenized equation in space and time with constant coefficients; and (3) then assume that 33 the homogenized equation is still valid even when the coefficients vary in space and the body is finite. 34 Roughly, our procedure assumes implicitly a separation of scales between the periodic microstructure and 35 the macroscopic heterogeneity. 36 To test and demonstrate the homogenized model, we turn to an important topic of current interest in the 37 community: to predict the scattering of waves at boundaries between different metamaterials or boundaries 38 between a metamaterial and a homogeneous material, e.g. [SW17, Wil20b, Wil20a, CGMP19]. We use 39 problems extracted from this body of work as examples on which to test our approach, and discuss this in 40 detail below. Our approach, in brief, is to identify the additional nonstandard continuity conditions that 41 arise at a surface of discontinuity due to the higher-order derivatives in the homogenized equations. These 42 higher-order continuity conditions then provide precisely the information required to uniquely solve for the 43 wave scattering at the boundary. A schematic view of this approach is shown in Figure1. 44 Prior Work on Scattering at Boundaries between Metamaterials and Homogeneous Materials. A 45 simplistic approach to model the scattering at a boundary might start from the usual assumption of infinite 46 periodicity to obtain the homogenized properties of the materials on either side, and then use these properties 47 in the nonperiodic setting with a boundary. However, as pointed out in [SW17], such an approach would 48 consider only the propagating waves and miss the contribution from the evanescent waves that are present 49 in the nonperiodic setting. They showed that if such homogenized models are applied to this problem, 50 the evanescent modes cannot be accounted for, and the classical continuity conditions – continuity of 51 displacement and traction – must be satisfied with propagating waves only; this leads to a violation of the 52 energy flux conservation. 3 53 [SW17] discuss an approach to deal with this problem. Their approach is to minimize the errors in 54 displacement continuity and traction continuity, subject to the conservation of energy, by considering only 55 the propagating modes. While simple and easy to use, it does not consider the microstructural information 56 in formulating this criterion, i.e., it does not account for the details of the evanescent modes. Further, it 57 can be difficult to apply this in a more general setting, for instance in a setting with complex geometry and 58 time-dependence where a decomposition into propagating modes is difficult. 59 A different “bottom-up” approach to this problem, and analogous problems, has been proposed by other 60 workers, e.g., [MMP18, CG20, CGMP19, PMM21, MM17, MM18, MPM19, GMOI19, MG18]. These 61 approaches rigorously account for the lack of periodicity, e.g., by introducing boundary-corrector functions 62 or using matched asymptotic expansions in 2-scale methods. Such methods typically require the coupled 63 solution of the cell problem – solving for the fast variable in a periodic unit cell – along with the effective 64 PDE describing the wave transmission, or the computation of the boundary-corrector functions. This 65 requirements make the approaches challenging and computationally expensive, and consequently difficult 66 to apply in a general setting. Further, these methods typically are restricted to a relatively limited frequency 67 band, making them difficult to apply to general problems with complex time-dependent loading. 68 The Proposed Approach.The approach that we propose aims to sit between the approach of [SW17] and the 69 rigorous homogenization approaches. Specifically, we aim for our homogenized models to be (1) applicable 70 to a broad range of frequencies, (2) applicable to general geometries and time-dependent loadings, (3) 71 computationally efficient and easy to compute with. 72 In the specific context of the insights provided by [SW17] on the importance of evanescent waves, our 73 homogenized models are able to capture the evanescent waves and the bandgap. Further, the higher-order 74 continuity conditions (Figure1) at the interface augment the standard continuity conditions, and provide 75 unique solutions that compare well with exact fine-scale calculations. 76 Emergence of Temporal Nonlocality with Multiple Bands. Existing dynamic homogenization techniques 77 can provide a homogenized equation for a specific band of the dispersion curve by expanding about the 78 frequency in that band, e.g. [GMOI19, CKP10, HMC16]. However, this typically does not provide a 79 single equation that covers several bands. In this work, we propose a single homogenized equation that is 80 applicable over a range of frequencies that covers multiple bands and the bandgaps. An important feature of 81 this homogenized equation is that it has higher derivatives in time, suggesting the emergence of nonlocality 82 in time due to homogenization as we increase the number of bands.
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